Simon Benninga. Financial Modelling 3-rd edition

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438 Chapter 16

Since the payoffs in the future are all nonnegative (with a positive

probability of being positive), it follows that the initial cash ﬂ ow from

the position must be negative:

213 2

CCC C

−−<⇒<

This proves the proposition. (Note that the assumption that

is made for convenience and does not affect the generality of the

argument.)

Without proof we state a similar proposition for puts:

proposition 6 (put price convexity) Consider three European puts, all

written on the same non-dividend-paying stock and with the same expi-

ration date T. We suppose that the exercise prices on the calls are X

, X

and X

, and denote the associated put prices by P

, P

, and P

. We further

assume that

. Then the put price is a convex function of

the exercise price:

Finally, we state the following proposition, whose proof involves only

a minor modiﬁ cation of the proof of Proposition 1:

proposition 7 (call option bounds with a known future dividend)

Consider a call with exercise price X and maturity date T. Suppose that

at some time t < T, the stock will, with certainty, pay a dividend D. Then

the lower bound on the call option price is given by

CSDeXe

rt rT

0≥−−

−−

max ( ),

Proof The proof involves only a minor modiﬁ cation of the proof of

Proposition 1.

439 An Introduction to Options

Today

Time t At Time T

Action Cash Flow

< XS

≥ X

Buy stock

−S

+D +S

Borrow the PV of the

dividend D

+De

−rT

−D

Borrow PV of X +Xe

−rT

−X −X

Write call

−(S

− X)

Total

−S

+ De

−rT

+ C

− X ≤ 0

This proves the proposition.

16.7 Summary

This chapter summarizes the basic deﬁ nitions and features of options. It

is, however, by no means an adequate introduction to these complex

securities for those with no preknowledge. To decipher the mysteries of

options, we recommend the introductory chapters of a good option

text.

Exercises

1. When you looked at the newspaper quotes for options on ABC stock, you saw that

a February call option with X = 37.5 is priced at 6.375, whereas the April call option

with the same exercise price is priced at 6. Can you devise an arbitrage out of these

prices? Do you have an explanation for the newspaper quotes?

2. An American call option is written on a stock whose price today is S = 50. The

exercise price of the call is X = 45.

a. If the call price is 2, explain how you would use arbitrage to make an immediate

proﬁ t.

b. If the option is exercisable at time T = 1 year and if the interest rate is 10%, what

is the minimum price of the option? Use Proposition 1.

3. A European call option is written on a stock whose current price S = 80. The exercise

price X = 80, the interest rate r = 8%, and the time to option exercise T = 1. The

stock is assumed to pay a dividend of 3 at time t =

. Use Proposition 7 to determine

the minimum price of the call option.

440 Chapter 16

4. A put with an exercise price of 50 has a price of 6, and a call on the same stock with

an exercise price of 60 has a price of 10. Both put and call have the same expiration

date.

a. On the same set of axes, draw the “proﬁ t” diagram for

•

One put bought and one call bought.

•

Two puts bought and one call bought.

•

Three puts bought and one call bought.

b. All three lines cross each other for the same value of S

. Derive this value.

5. Consider the following two calls:

•

Both calls are written on shares of ABC Corporation, whose current share price

is $100. ABC does not pay any dividends.

•

Both calls have one year to maturity.

•

One call has X

= 90 and has price of 30; the second call has X

= 100 and has

price of 20.

•

The riskless, continuously compounded interest rate is 10 percent.

By designing a spread position (i.e., buying one call and writing another), show that

the difference between the two call prices is too large and that a riskless arbitrage

exists.

6. A share of ABC Corporation sells for $95. A call on the share with exercise price

$90 sells for $8.

a. Graph the proﬁ t pattern from buying one share and one call on the share.

b. Graph the proﬁ t pattern from buying one share and two calls.

c. Consider the proﬁ t pattern from buying one share and N calls. At which share

price do all the proﬁ t lines cross?

7. A European call with a maturity of six months and exercise price X = 80, written

on a stock whose current price is 85, is selling for $12.00; a European put written

on the same stock with the same maturity and with the same exercise price is selling

for $5.00. If the annual interest rate (continuously compounded) is 10 percent, con-

struct an arbitrage from this situation.

8. Prove Proposition 6. Then solve the following problem:

Three puts on shares of XYZ with the same expiration date are selling at the fol-

lowing prices:

Exercise price 40: 6

Exercise price 50: 4

Exercise price 60: 1

Show an arbitrage strategy that allows you to proﬁ t from these prices and prove

that it works.

9. The current stock price of ABC Corporation is 50. Prices for six-month calls on

ABC are given in the following table:

441 An Introduction to Options

Call Price

40 16.5

50 9.5

60 4.5

70 2

Draw a proﬁ t diagram of the following strategy: Buy one 40 call, write two 50 calls,

buy one 60 call, and write two 70 calls.

10. Consider the following option strategy, which consists only of calls:

Exercise Price Bought/Written? Number? Price per Call Option

20 1 written 45

30 2 bought 33

40 1 written 22

50 1 bought 18

60 2 written 17

70 1 bought 16

a. Draw the proﬁ t diagram for this strategy.

b. The prices given include one violation of an arbitrage condition. Identify this

violation and explain.

11. A share of Formila Corporation is currently trading at $38.50, and a one-year call

option on Formila with X = $40 is trading at $3. The risk-free interest rate is

4.5%.

a. What should be the price of a one-year put option on the stock with X = $40?

Why?

b. If the price of a put is $2, construct an arbitrage strategy.

c. If the price of a put is $4, construct an arbitrage strategy.

The Binomial Option-Pricing Model

17.1 Overview

Next to the Black-Scholes model (discussed in Chapter 19), the binomial

option-pricing model is the most widely used option-pricing model. It

has many advantages: It is a simple model, which—in addition to giving

many insights into option pricing—is easily programmed and adapted to

numerous, and often quite complicated, option-pricing problems. When

extended to many periods, the binomial model becomes one of the most

powerful ways of valuing securities like options whose payoffs are con-

tingent on the market prices of other assets.

The binomial model depends on using state prices to compute the

values of risky assets. When the state-pricing principles underlying the

model are understood, we gain deeper insight into the economics of

contingent asset pricing. In this chapter, which illustrates the simple uses

of the binomial model, we devote a considerable amount of space to

deriving and using state prices. In Chapters 22 and 23 we return to the

binomial model and use it in the Monte Carlo pricing of contingent

securities.

17.2 Two-Date Binomial Pricing

To illustrate the use of the binomial model, we start with the following

very simple example:

•

There is one period and two dates; date 0 represents today and date 1

is one year from now.

•

There are two “fundamental” assets: a stock and a bond. There is also

a derivative asset, a call option written on the stock.

•

The stock price today is $50. At date 1 it will either go up by 10 percent

or go down by 3 percent.

•

The one-period interest rate is 6 percent.

•

The call option matures at date 1 and has exercise price X = $50.

Here is a picture from a spreadsheet that incorporates this model.

Notice that in cells B2, B3, and B6 we have used values for 1 plus the

10 percent up move, 1 plus the −3 percent down move, and 1 plus the

444 Chapter 17

6 percent interest. We use capital letters U, D, and R to denote these

values.

ABCD EFGHI J

Up, U 1.10

Down, D 0.97

Initial stock price 50.00

Interest rate, R 1.06

Exercise price 50.00

Stock price Bond price

55.00 <-- =$B$11*B2 1.06 <-- =$G$11*$B$6

50.00 1.00

48.50 <-- =$B$11*B3 1.06 <-- =$G$11*$B$6

Call option

5.00 <-- =MAX(D10-$B$7,0)

???

0.00 <-- =MAX(D12-$B$7,0)

BINOMIAL OPTION PRICING IN A ONE-PERIOD MODEL

We wish to price the call option. We do so by showing that there is a

combination of the bonds and stocks that exactly replicates the call option’s

payoffs. To show this fact, we use some basic linear algebra; suppose we

ﬁ nd A shares of the stock and B bonds such that

55 1 06 5AB+=.

48 5 1 06 0..AB+=

This system of equations solves to give

A =

−

55 48 5

0 7692

−

=−

0485

106

35 1959

Thus purchasing 0.77 of a share of the stock and borrowing $35.20 at 6

percent for one period will give payoffs of $5 if the stock price goes up

and $0 if the stock price goes down—the payoffs of the call option. It

follows that the price of the option must be equal to the cost of replicat-

ing its payoffs; that is,

1. Should there be a need to distinguish between 1.10 (1 plus the 10 percent up move of

the stock) and 10 percent (the up move itself), we will use U for the former and low-

ercase u for the latter.

445 The Binomial Option-Pricing Model

Call option price =∗− =0 7692 50 35 1959 3 2656.$$. $.

This logic is called “pricing by arbitrage”: If two assets or sets of assets

(in our case—the call option and the portfolio of 0.77 of the stock and

−$35.20 of the bonds) have the same payoffs, they must have the same

market price.

ABCD EFG

Solving the portfolio problem: A shares + B bonds combine to give option payoffs

A 0.7692 <-- =D15/(D10-D12)

B -35.1959 <-- =-D12*B20/B6

Call price 3.2656 <-- =B20*B5+B21

In succeeding sections we show that this simple arbitrage argument

can be extended to multiple periods. But in the meantime we conﬁ ne

ourselves in the next section to generalizing the logic.

17.3 State Prices

There is actually a simpler (and more general) way to solve this problem:

Viewed from today, there are only two possibilities for next period:

Either the stock price goes up or it goes down. Think about the market

determining a price q

for $1 in the “up” state of the world and a price

for $1 in the “down” state of the world. Then both the bond and the

stock have to be priced using these state prices:

qSUq SDS qUqD

UD UD

∗∗ + ∗∗ = ⇒ + =1

qRqR

∗+ ∗=1

The state prices are thus an illustration of the linear pricing principle:

If the stock price can move up in one period by a factor U and down by

a factor R, and if 1 plus the one-period interest rate is R, then any other

asset will be priced by discounting its payoff in the “up” state by q

and

by discounting its payoff in the “down” state by q

The two preceding equations solve to give

RU D

−

−() ()

In our case these state prices are given by

446 Chapter 17

CBA

Up, U 1.10

Down, D 0.97

Interest rate, R 1.06

State prices

0.6531 <-- =(B4-B3)/(B4*(B2-B3))

0.2903 <-- =(B2-B4)/(B4*(B2-B3))

Pricing the stock: 1 = q

*U+q

*D?

1 <-- =B7*B2+B8*B3

Pricing the bond: 1/R = q

1.06 <-- =1/(B7+B8)

Check: Confirm that state prices actually price the stock and the bond

DERIVING THE STATE PRICES

In rows 11 and 12 we check that the state prices indeed give back the

interest rate and the stock price.

We can now use the state prices to price the call and the put on the

stock and also to establish that put-call parity holds. The call and the put

options should be priced by

Cq SUX q SDX

=∗−

()

+∗−

()

max , max ,00

Pq XSU q XSD

=−∗

()

+−∗

()

max , max ,00

or, if priced by put-call parity,

PCPVX S=+ −()

We can present these relations in a spreadsheet:

CBA

Up, U 1.10

Down, D 0.97

Interest rate, R 1.06

Initial stock price, S 50.00

Option exercise price, X 50.00

State prices

0.6531 <-- =(B4-B3)/(B4*(B2-B3))

0.2903 <-- =(B2-B4)/(B4*(B2-B3))

Pricing the call and the put

Call price 3.2656 <-- =B9*MAX(B5*B2-B6,0)+B10*MAX(B5*B3-B6,0)

Put price 0.4354 <-- =B9*MAX(B6-B5*B2,0)+B10*MAX(B6-B5*B3,0)

Put-call parity

Stock + put 50.4354 <-- =B5+B14

Call + PV(X) 50.4354 <-- =B13+B6/B4

BINOMIAL OPTION PRICING WITH STATE PRICES IN A ONE-PERIOD

(TWO-DATE) MODEL

Note about PV(X) in put-call parity:

In the continuous-time framework (the standard Black-Scholes framework), PV(X) = X*Exp(-r*T). Because

the framework here is discrete time, PV(X) is also discrete-time: PV(X)=X/(1+r)=X/R.

447 The Binomial Option-Pricing Model

The formulas we use (with S = 50, X = 50, U = 1.10, D = 0.97, R = 1.06)

are

Cq SUX q SDX

=∗−+∗−

()

=∗+∗

max , max ,() ..

0 0 0 6531 5 0 2903 0

3 2657

for the call, and

Pq XSU q XSD

=−∗+−∗

=∗ −+

max , max ,

max ,

()()

.().

0 6531 50 55 0 0 2903∗∗−=max ,(.).50 48 5 0 0 4354

for the put. As expected—the put-call parity theorem holds for this par-

ticular put and call (cells B17:B18):

PS C

+= + =+ = +0 4354 50 3 27

106

17.3.1 State Prices or Risk-Neutral Prices?

Multiplying the state prices by 1 plus the interest rate R gives the risk-

neutral prices: π

= q

R, π

= q

R. The risk-neutral prices look like a

probability distribution of the states, since they sum to 1:

ππ

UDU D

qR qR

RU D

R+= + =

−

()()

Furthermore, there is a fundamental equivalence of pricing by the risk-

neutral prices and pricing by the state prices. Suppose an asset has state-

dependent payoffs X

in the “up” state and X

in the “down” state of a

two-date model. Then the date-0 price of the asset using the state prices

is q

+ q

, and the date-0 price of the asset using the risk-neutral

prices is the discounted expected asset payoff, where the expectation is

computed using the risk-neutral prices as if they are the actual state

probabilities:

ππ

UU DD

“”Expected asset payoff using risk-neutral pricees

qX qX

UU DD

These two computations are, of course equal. This author prefers to use

state prices, but many researchers are more comfortable using the pseu-

doprobabilities of the risk-neutral prices and then discounting the

“expected” payoffs.